where the trace operation is applied to all the bath degrees of freedom, thus reducing the dimensionality of the problem to the number of degrees of freedom of \(S\) . \(\hat{V}(t,t_{0})\)  is an effective propagator that acts on the sole \(S\)  sub-space. It turns out that an expression for \(\hat{V}(t,t_{0})\)  that verifies Eq. REF  exists and that it can be derived with the formalism of projection operators. This approach can be used to derive either the Nakajima-Zwanzig or time-convolutionless equations[1]}. The latter provides an exact time-local equation of motion for \(\hat{\rho }_{S}(t)\)  in the form
\(\frac{d\hat{\rho }_{S}(t)}{dt}=\hat{K}(t,t_{0})\hat{\rho }_{S}(t)\:.\) 
